The analysis and application of Optimal Transport for Seismic Inversion
Yunan Yang, University of Texas at Austin
Optimal transport has become a well-developed topic in the mathematical analysis. Due to their ability to incorporate differences in both signal intensity and spatial information, the related Wasserstein metrics have been adopted in a variety of applications, including seismic inversion. Full Waveform Inversion (FWI) is a PDE-constrained optimization in which the variable velocity in a forward wave equation is adjusted such that the solution matches measured data on the surface. $L^2$ norm is the conventional objective function measuring the difference between simulated and measured data, but it often results in the minimization trapped in local minima. One way to mitigate this is by selecting another misfit function with better convexity, and we proposed using the quadratic Wasserstein metric ($W_2$). The optimal map defining $W_2$ can be computed by quicksort (trace-by-trace comparison) or solving a Monge-Amp\`ere equation (global comparison). Theorems pointing to the advantages of using optimal transport over $L^2$ norm will be discussed, and some large-scale computational examples will be presented. There is an interesting question about how to adapt datasets that are not naturally nonnegative and mass balanced into optimal transport theory. It is joint work with Bjorn Engquist (UT Austin) and Brittany Froese Hamfeldt (NJIT).